Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

• J. B. Bost ^[1] ; A. Connes ^[1]
  1. [1] Institut des Hautes Études Scientifiques

     Institut des Hautes Études Scientifiques

     Arrondissement de Palaiseau, Francia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 1, Nº. 3, 1995, págs. 411-457
• Idioma: inglés
• DOI: 10.1007/bf01589495
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• Resumen

  • In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann ζ function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings ℤ⊂ℚ provides the desiredC*-dynamical system, which admits the ζ function as partition function and the Galois group Gal(ℚcycl/ℚ) of the cyclotomic extension ℚcycl of ℚ as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β=1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).